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Combining Probabilities
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The basic principle of probability can be stated as follows: If one event has c possible outcomes and a second event has d possible outcomes, then there are cd possible outcomes of the two events. From this principle, we obtain three rules that concern us as geneticists. To understand these rules of probability requires a few de nitions. Mutually exclusive events are events in which the occurrence of one possibility excludes the occurrence of the other possibilities. In the throwing of a die, for example, only one face can land up. Thus, if it comes up a four, it precludes the possibility of any of the other faces. Similarly, a blue-eyed daughter is mutually exclusive of a brown-eyed son or any other combination of gender and eye color. Independent events, however, are events whose outcomes do not in uence one another. For example, if two dice are thrown, the face
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Types of Probabilities
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The probability (P) that an event will occur is the number of favorable cases (a) divided by the total number of possible cases (n): P a/n
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The probability can be determined either by observation (empirical) or by the nature of the event (theoretical). For example, we observe that about one child in ten thousand is born with phenylketonuria. Therefore, the
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II. Mendelism and the Chromosomal Theory
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4. Probability and Statistics
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Probability and Statistics
value of one die is not able to affect the face value of the other; they are thus independent of each other. Similarly, the gender of one child in a family is generally independent of the gender of the children who have come before or might come after. Finally, unordered events are events whose probability of outcome does not depend on the order in which the events occur; the probabilities combine both mutual exclusivity and independence. For example, when two dice (one red, one green) are thrown at the same time, we generally do not specify which die has which value; a seven can occur whether the green die is the four or the red die is the four. Similarly, the probability that a family of several children will have two boys and one girl is the same irrespective of their birth order. If the family has two boys and one girl, it does not matter whether the daughter is born rst, second, or third. In general, probabilities differ depending on whether order is speci ed. With these de nitions in mind, let us look at three rules of probability that affect genetics.
USE OF RULES
There are several ways to calculate the probability just asked for. To put the problem in the form for rule 3 is the quickest method, but this problem can also be solved by using a combination of rules 1 and 2. For each penny, the probability of getting a head ( H) or a tail ( T ) is for H: P for T: Q 1/2 1/2
Tossing the pennies one at a time, it is possible to get a head and a tail in two ways: rst head, then tail (HT) or rst tail, then head (TH) Within a sequence (HT or TH), the probabilities apply to independent events. Thus, the probability for any one of the two sequences involves the product rule (rule 2): 1/2 1/2 1/4 for HT or TH
1. Sum Rule
When events are mutually exclusive, the sum rule is used: The probability that one of several mutually exclusive events will occur is the sum of the probabilities of the individual events. This is also known as the either-or rule. For example, what is the probability, when we throw a die, of its showing either a four or a six According to the sum rule, P 1/6 1/6 2/6 1/3
The two sequences (HT or TH) are mutually exclusive. Thus, the probability of getting either of the two sequences involves the sum rule (rule 1): 1/4 1/4 1/2
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